Question

eh is a diameter of ⊙d. the measure of $widehat{ef}$ is $(10x + 8)^{circ}$ and the measure of $widehat{gh}$ is $(11x)^{circ}$. determine the values. the value of x is the measure of $widehat{ef}$ is degrees. the measure of $widehat{gh}$ is degrees.

Explanation:

Step1: Set up equation

Since $EH$ is a diameter, the sum of the measures of $\overset{\frown}{EF}$, $\overset{\frown}{FG}$ and $\overset{\frown}{GH}$ is $180^{\circ}$. Given $\overset{\frown}{FG}=67^{\circ}$, $\overset{\frown}{EF}=(10x + 8)^{\circ}$ and $\overset{\frown}{GH}=(11x)^{\circ}$, we have the equation $(10x + 8)+67+11x=180$.

Step2: Simplify the left - hand side

Combine like terms: $10x+11x + 8+67=21x + 75$. So, $21x+75 = 180$.

Step3: Solve for $x$

Subtract 75 from both sides: $21x=180 - 75=105$. Then divide both sides by 21: $x=\frac{105}{21}=5$.

Step4: Find the measure of $\overset{\frown}{EF}$

Substitute $x = 5$ into the expression for $\overset{\frown}{EF}$: $\overset{\frown}{EF}=10x + 8=10\times5+8=50 + 8=58^{\circ}$.

Step5: Find the measure of $\overset{\frown}{GH}$

Substitute $x = 5$ into the expression for $\overset{\frown}{GH}$: $\overset{\frown}{GH}=11x=11\times5 = 55^{\circ}$.

Answer:

The value of $x$ is $5$.
The measure of $\overset{\frown}{EF}$ is $58$ degrees.
The measure of $\overset{\frown}{GH}$ is $55$ degrees.